Abstract

We present competitive and uncompetitive drug–drug interaction (DDI) with target mediated drug disposition (TMDD) equations and investigate their pharmacokinetic DDI properties. For application of TMDD models, quasi-equilibrium (QE) or quasi-steady state (QSS) approximations are necessary to reduce the number of parameters. To realize those approximations of DDI TMDD models, we derive an ordinary differential equation (ODE) representation formulated in free concentration and free receptor variables. This ODE formulation can be straightforward implemented in typical PKPD software without solving any non-linear equation system arising from the QE or QSS approximation of the rapid binding assumptions. This manuscript is the second in a series to introduce and investigate DDI TMDD models and to apply the QE or QSS approximation.

The next step is to differentiate Eq. (67) and to express \(\frac{d}{dt} C_{totA}, \frac{d}{dt} C_{totB}, \frac{d}{dt} R_{tot}\) appearing at the left hand side of Eqs. (54)–(56) in terms of \(C_A, C_B\) and R and their derivatives. Using Eqs. (51)–(53) we can calculate from Eqs. (54)–(56)

Appendix 2: QE approximation

The QE approximation is based on the theory of Fenichel [14] which allows a specific selection of the rates to be accelerated.

Competitive

To justify the QE approximation we increase the binding rates \(k_{onX},k_{offX}\), where \(X \in \{A,B\}\), by replacing with \(\frac{1}{\varepsilon } k_{onX}\), \(\frac{1}{\varepsilon } k_{offX}\) with \(\varepsilon > 0\) small in Eqs. (54)–(58). Since the new constants are much larger this can be regarded as rapid binding and we obtain

compare the lines 113–128 for the competitive and the lines 221–239 for the uncompetitive case. The variables \(H_1\),...,\(H_3\) correspond to DADT(1), ..., DADT(3) in NONMEM and XP(1), ..., XP(3) in ADAPT 5.

The lines of the code are numbered for referencing but are not part of the code implementation.

NONMEM control stream for competitive DDI TMDD

The $DES block of the control stream is presented. Additionally, the first lines of the data file is shown to present the IV infusion mechanism. The full control stream is available in the supplemental material.

101: $DES

102: EPSILON = 1e-4

103: ; Dose at T1 = 0

104: INA = 0

105: INB = 0

106: IF (T.GE.0.AND.T.LE.0+EPSILON) THEN

107: INA = 100*EPSILON**(−1)

108: INB = 100*EPSILON**(-1)

109: ENDIF

110: CA = A(1)/V

111: CB = A(2)/V

112: R = A(3)

113: DET = R**2+CA*KDB+CB*KDA+CA*R+CB*R+KDA*KDB+KDA*R+KDB*R

114: G1 = INA - KELA*CA - (KINTA*CA*R)/KDA

115: G2 = INB - KELB*CB - (KINTB*CB*R)/KDB

116: G3 = KSYN-KDEG*R-(KINTA*CA*R)/KDA-(KINTB*CB*R)/KDB

117: M11 = (1/DET)*(DET - R*(R+CB+KDB))

118: M12 = (1/DET)*(CA*R)

119: M13 = (1/DET)*(-CA*(R+KDB))

120: M21 = (1/DET)*(CB*R)

121: M22 = (1/DET)*(DET - R*(R+CA+KDA))

122: M23 = (1/DET)*(-CB*(R+KDA))

123: M31 = (1/DET)*(-R*(R+KDB))

124: M32 = (1/DET)*(-R*(R+KDA))

125: M33 = (1/DET)*(DET-CA*(R+KDB)-CB*(R+KDA))

126: DADT(1) = M11*G1 + M12*G2 + M13*G3

127: DADT(2) = M21*G1 + M22*G2 + M23*G3

128: DADT(3) = M31*G1 + M32*G2 + M33*G3

The first lines of the data file are:

150: #ID TIME TYPE DV MDV

151: 1 0 1 . 1

152: 1 0 2 . 1

153: 1 0.0001 1 . 1

154: 1 0.0001 2 . 1

155: 1 2 1 32.9432 0

156: 1 2 2 28.3621 0

ADAPT 5 source code for uncompetitive DDI TMDD

The subroutine DIFFEQ is presented. For full source code see supplemental material.

Properties of the original DDI TMDD models. Competitive: In panels a and b the single drug profiles of drugs A and B (blue dotted lines) are compared with the competitive model (black solid line) Eqs. (1)–(9) for \(dose_A = 100\), \(dose_B = 100\) and no baseline \(C_A^0 = C_B^0 = 0\). In panels c and d, the effect of the administration of one drug only on a present concentration of the other drug is shown by two examples: (i) drugs A and B are in baseline \(C_A^0 = C_B^0 = 1\), and one administration at time \(t = 12\) of drug B with \(dose_B = 100\) (red dashed lines) causes an increase of drug A concentration, (ii) drug A is administered with \(dose_A = 100\) at \(t = 0\) and drug B administered with \(dose_B = 100\) at \(t = 0\) and additionally at \(t = 12\) (black solid lines), and causes an increase of drug A concentration. Uncompetitive: In panels e and f the single drug profiles of drugs A and B (blue dotted lines) are compared with the uncompetitive model (black solid lines) Eqs. (23)–(33) for \(dose_A = 100\), \(dose_B = 100\) and no baseline \(C_A^0 = C_B^0 = 0\). In panels g and h drugs A and B are in baseline \(C_A^0 = C_B^0 = 1\) with one administration at time \(t=5\) of drug A with \(dose_A = 100\) and non of drug B (red dashed line) and the other way around (black solid lines) (Color figure online)

Visualization of the QE approximation. Competitive: In panels a and b concentration profiles from the original formulation (red dashed lines) Eqs. (1)–(9) and the approximation of the QE formulation Eqs. (14)–(18) (black solid lines) are shown for escalating doses of \(dose_A = dose_B = 10, 100, 1000\) at \(t=0\) and no baseline \(C_A^0 = C_B^0 = 0\). In panels c and d the effect of one drug administration on the present concentration of the other drug is shown. The original (red dashed lines) and QE approximation (black solid lines) profiles with a baseline \(C_A^0 = C_B^0 = 1\) are shown for a dose of \(doseA = 1000\) at \(t=0\). The \(k_{onB}\) and \(k_{offB}\) are multiplied by the factors 0.1, 1 and 10 in such a way that \(K_{DB}\) stays the same to show the convergence of the original formulation towards the QE approximation. Uncompetitive: In panels e and f concentration profiles from the original formulation (red dashed lines) Eqs. (23)–(33) and the approximation of the QE formulation Eqs. (43)–(48) (black solid lines) are shown for escalating doses of \(dose_A = dose_B = 10, 100, 1000\) at \(t=0\) and no baseline \(C_A^0 = C_B^0 = 0\). In panels g and h original (red dashed lines) and QE approximation (black solid lines) profiles with a baseline \(C_A^0 = C_B^0 = 1\) are shown where for drug A is administered with a dose of \(dose_A = 100\) at \(t=24\). The \(k_{onA}\) and \(k_{offA}\) are multiplied by the factors 0.1, 1 and 10 in such a way that \(K_{DA}\) stays the same (Color figure online)

Visualization of plasma concentration versus time data fitting from the original formulation with the QE approximation: Fit (solid lines) of the QE approximation of the competitive Eqs. (14)–(18) in NONMEM (panels a and b) and the uncompetitive DDI TMDD model Eqs. (43)–(48) in ADAPT 5 (panels c and d) in ODE formulation with an IV short infusion. Data (crosses) were produced with the original formulations Eqs. (1)–(9) and Eqs. (23)–(33)